Timeline for A quadratic optimization problem involving Brownian motion
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 19, 2022 at 6:38 | comment | added | megaproba | When $f \in H^1(0,1)$, using the integration by part proposed by ofer zeitouni in the comment above, you can solve the optimization problem $\omega$ by $\omega$ on the canonical space of the Brownian motion. The Wiener measure becomes an example amongst others. | |
| Jun 18, 2022 at 14:27 | comment | added | ofer zeitouni | That's a different question. Note that if $f$ is differentiable, the stochastic integral becomes $B(1) f(1)-\int_0^1 f'(s) B_s ds$. If you put a constraint on (say) the $H^1$ norm, you can optimize, it is a standard optimization problem (but the final f will not be adapted). | |
| Jun 18, 2022 at 3:04 | comment | added | Zuofeng Shang | Thanks for your excellent example. Motivated by your example, the infinite minimum is probably due to the discontinuity of f. If we require f being sufficiently smooth, shall the minimum exist? For example, if we require f being constant, the minimum is -(B_1-B_0)^2 achieved at f=B_1-B_0. I am not sure if it is possible to find a weakest smooth condition on f such that the minimum finitely exists. | |
| Jun 17, 2022 at 10:33 | history | edited | Nawaf Bou-Rabee | CC BY-SA 4.0 |
fixed minor typos
|
| Jun 17, 2022 at 8:49 | history | answered | ofer zeitouni | CC BY-SA 4.0 |